On the Gram Matrices of Systems of Uniformly Bounded Functions

Let $A_N$, $N=1,2,\dots $, be the set of the Gram matrices of systems $\{e_j\}_{j=1}^N$ formed by vectors $e_j$ of a Hilbert space $H$ with norms $\|e_j\|_H\le 1$, $j=1,\dots ,N$. Let $B_N(K)$ be the set of the Gram matrices of systems $\{f_j\}_{j=1}^N$ formed by functions $f_j\in L^\infty (0,1)$ with $\|f_j\|_{L^\infty (0,1)}\le K$, $j=1,\dots ,N$. It is shown that, for any $K$, the set $B_N(K)$ is narrower than $A_N$ as $N\to \infty$. More precisely, it is proved that not every matrix $A$ in $A_N$ can be represented as $A=B+\Delta $, where $B\in B_N(K)$ and $\Delta $ is a diagonal matrix.